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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习5.1-5.3 - 非负可测函数的勒贝格积分 }
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\date{2024 年 5 月 13 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $E\subseteq\mathbb{R}^n$ 是可测集，什么是 $E$ 上的一个非负简单函数？可测集 $E$ 上的非负简单函数的勒贝格积分是怎么定义的？计算狄利克雷函数的勒贝格积分。

\vspace{0.1cm}

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\item  %Problem 02、S5.2定理1
设 $E$ 是可测集，设 $\varphi(x)$ 是 $E$ 上的一个非负简单函数。证明： 
\begin{enumerate}
\item  对任意非负实数 $c$, 有 $$ \int_E c\varphi(x)dx = c\int_E \varphi(x)dx. $$ 
\item  设 $A,B$ 是 $E$ 的两个不相交的可测子集，则 $$ \int_{A\cup B} \varphi(x)dx = \int_A \varphi(x)dx + \int_B \varphi(x)dx. $$ 
\item  设 $\{A_n\}$ 是 $E$ 的一列可测子集，满足 $A_1\subset A_2\subset A_3\subset\cdots\subset A_n\subset\cdots$ 与 $\cup_{n=1}^{\infty} A_n = E$, 则有 
$$\lim\limits_{n\to\infty}\int_{A_n} \varphi(x)dx = \int_E \varphi(x)dx. $$ 
\end{enumerate}

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%\item  %Problem 03、S5.2定理2
%设 $E$ 是可测集，设 $\varphi(x)$ 和 $\psi(x)$ 是 $E$ 上的非负简单函数。证明： 
%\begin{enumerate}
%\item  $$\int_E [\varphi(x)+\psi(x)]dx = \int_E \varphi(x)dx + \int_E \psi(x)dx. $$ 
%\item  对任意非负实数 $\alpha,\beta$, 有
%$$\int_E [\alpha\varphi(x) + \beta\psi(x)]dx = \alpha\int_E \varphi(x)dx + \beta\int_E \psi(x)dx. $$
%\end{enumerate}
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%\vspace{0.1cm}

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\item  %Problem 04
可测集 $E$ 上的非负可测函数 $f(x)$ 的勒贝格积分是怎么定义的？
什么时候称 $f(x)$ 在 $E$ 上是勒贝格可积的？

\vspace{0.1cm}

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\item  %Problem 05、S5.3定理1
设 $E\subseteq\mathbb{R}^n$ 是可测集，设 $f(x)$ 是 $E$ 上的一个非负可测函数，证明： 
\begin{enumerate}
\item  若 $m(E)=0$, 则 $$\int_Ef(x)dx=0.$$  
\item  若 $\int_Ef(x)dx=0,$ 则 $f(x)=0$ 在 $E$ 上几乎处处成立。 
\item  若 $\int_Ef(x)dx<\infty,$ 则 $0\le f(x)<\infty$ 在 $E$ 上几乎处处成立。 
\item  设 $A,B$ 是 $E$ 的两个不相交的可测子集，则 $$ \int_{A\cup B} f(x)dx = \int_A f(x)dx + \int_B f(x)dx. $$ 
\end{enumerate}

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%\item  %Problem 06、S5.3定理2
%设 $f(x)$ 与 $g(x)$ 是可测集 $E\subset\mathbb{R}^n$ 上的非负可测函数，证明： 
%\begin{enumerate}
%
%\item  若 $f(x)\le g(x)$ 在 $E$ 上几乎处处成立，则 $$\int_Ef(x)dx \le \int_Eg(x).$$ 
%这时，若 $g(x)$ 在 $E$ 上勒贝格可积，则 $f(x)$ 在 $E$ 上也勒贝格可积。  
%
%\item  若 $f(x)= g(x)$ 在 $E$ 上几乎处处成立，则 $$\int_Ef(x)dx = \int_Eg(x).$$  
%特别地，若 $f(x)$ 在 $E$ 上几乎处处为零，则 $$\int_Ef(x)dx = 0. $$ 
%
%\end{enumerate}
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\item  %Problem 07、S5.3定理3
（莱维定理）设 $E\subseteq\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上一列非负可测函数，设对任意 $x\in E$, 对任意正整数 $n$, 都有 $f_n(x)\le f_{n+1}(x)$. 记 $f(x) = \lim\limits_{n\to\infty} f_n(x)$, 则有 
$$\lim\limits_{n\to\infty}\int_E f_n(x)dx = \int_E f(x)dx. $$ 

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%\item  %Problem 08、S5.3定理4
%设 $E\subset\mathbb{R}^n$ 是可测集，设 $f(x)$ 和 $g(x)$ 是 $E$ 上的非负可测函数， 设 $\alpha$ 和 $\beta$ 都是非负实数，则有 $$\int_E [\alpha f(x) + \beta g(x)]dx = \alpha\int_E f(x)dx + \beta\int_E g(x)dx. $$
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\item  %Problem 09、S5.3定理5
（逐项积分定理）设 $E\subseteq\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上一列非负可测函数，则有
$$\int_E \left( \sum\limits_{n=1}^{\infty} f_n(x) \right)dx = \sum\limits_{n=1}^{\infty} \int_E f_n(x)dx. $$ 

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\item  %Problem 10、S5.3定理6
（法图引理）设 $E\subseteq\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上一列非负可测函数，则有
$$ \int_E \varliminf_{n\to\infty} f_n(x)dx \le \varliminf_{n\to\infty} \int_E f_n(x)dx. $$ 
举例说明等号不一定成立。

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\item  %Problem 11、S5习题1
设在康托尔集 $P$ 上定义函数 $f(x)=0$, 而在 $P$ 的余集中长为 $3^{-n}$ 的构成区间上定义为 
$n$, $(n=1,2,\cdots)$, 证明 $f(x)$ 可积，并求出积分值。

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\end{enumerate}


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\end{document}

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